Problem: Simplify the product \[\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12} \dotsm \frac{4n+4}{4n} \dotsm \frac{2008}{2004}.\]
Note that the 8 in the first fraction cancels with the 8 in the second fraction, the 12 in the second fraction cancels with the 12 in the third fraction, and so on.  We are then left with $\frac{2008}{4} = \boxed{502}.$